A119786 Numerator of the product of the n-th triangular number and the n-th harmonic number.

1, 9, 11, 125, 137, 1029, 363, 6849, 7129, 81191, 83711, 1118273, 1145993, 1171733, 1195757, 41421503, 42142223, 813635157, 275295799, 279175675, 56574159, 439143531, 1332950097, 33695573875, 34052522467, 309561680403, 312536252003, 9146733078187

Offset

1,2

Comments

Also numerator of the sum of all matrix elements of n X n matrix M[i,j] = i/j, i,j=1..n.

p^3 divides a(p-1) for prime p>3, p^3 divides a(p^2-1) for prime p>3, p^3 divides a(p^3-1) for prime p>3, p^3 divides a(p^4-1) for prime p>3, ...

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

a(n) = numerator[Sum[i,{i, 1, n}] * Sum[1/j,{j, 1, n}]] = numerator[n(n+1)/2 * Sum[1/i,{i, 1, n}]] = numerator[A000217(n) * (A001008(n)/A002805(n))]. Also a(n) = numerator[Sum[Sum[i/j,{i, 1, n}],{j, 1, n}]].

Maple

a:= n-> numer(add(add(i/j, j=1..n), i=1..n)): seq(a(n), n=1..30); # Zerinvary Lajos, Jun 14 2007
# second Maple program:
h:= proc(n) h(n):= 1/n +`if`(n=1, 0, h(n-1)) end:
t:= proc(n) t(n):= n +`if`(n=1, 0, t(n-1)) end:
a:= n-> numer(h(n)*t(n)):
seq(a(n), n=1..30); # Alois P. Heinz, May 24 2013

Mathematica

Numerator[Table[n(n+1)/2*Sum[1/i, {i, 1, n}], {n, 1, 50}]]. Numerator[Table[Sum[Sum[i/j, {i, 1, n}], {j, 1, n}], {n, 1, 50}]].
Table[(n(n+1))/2 HarmonicNumber[n], {n, 30}]//Numerator (* Harvey P. Dale, May 06 2018 *)

Crossrefs

Cf. A000217, A001008, A002805.

Sequence in context: A365221 A027727 A019328 * A147429 A147461 A146366

Adjacent sequences: A119783 A119784 A119785 * A119787 A119788 A119789

Keyword

frac,nonn

Author

Alexander Adamchuk, Jun 25 2006, Jul 12 2006

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

Status

approved